#394373

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2595 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,4)(5,6)(7,8)(9,10,15,21)(11,23,26,36)(12,28,31,18)(13,33,24,39)(14,22,16,20)(17,27,35,25)(19,29,40,34)(30,37,32,38);(1,2,3,4)(9,11,14,25,15,26,16,27)(10,17,20,23,21,35,22,36)(12,29,30,24,31,34,32,13)(18,33,37,19,28,39,38,40);(6,8)(9,12)(10,18)(11,24)(13,26)(14,32)(15,31)(16,30)(17,19)(20,38)(21,28)(22,37)(23,33)(25,29)(27,34)(35,40)(36,39);(9,13,15,24)(10,19,21,40)(11,12,26,31)(14,29,16,34)(17,28,35,18)(20,39,22,33)(23,38,36,37)(25,30,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^c=ei,g^c=gi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301569);